| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,0,1,2,2,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.1271'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+54t^5+64t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1271'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 928*K1**2*K2**3 - 3536*K1**2*K2**2 - 64*K1**2*K2*K4 + 4544*K1**2*K2 - 3448*K1**2 + 320*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4176*K1*K2*K3 + 416*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1560*K2**4 - 656*K2**2*K3**2 - 8*K2**2*K4**2 + 1648*K2**2*K4 - 2192*K2**2 + 432*K2*K3*K5 - 1120*K3**2 - 326*K4**2 - 72*K5**2 + 2436 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1271'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16997', 'vk6.17240', 'vk6.20530', 'vk6.21928', 'vk6.23402', 'vk6.23711', 'vk6.27985', 'vk6.29452', 'vk6.35469', 'vk6.35916', 'vk6.39389', 'vk6.41580', 'vk6.42903', 'vk6.43204', 'vk6.45968', 'vk6.47643', 'vk6.55172', 'vk6.55418', 'vk6.57393', 'vk6.58567', 'vk6.59552', 'vk6.59892', 'vk6.62060', 'vk6.63047', 'vk6.64976', 'vk6.65186', 'vk6.66941', 'vk6.67798', 'vk6.68266', 'vk6.68422', 'vk6.69552', 'vk6.70250'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3U1O4O5U3U2O6U4U6U5 |
| R3 orbit | {'O1O2O3U1O4O5U3U2O6U4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U6O5U2U1O4O6U3 |
| Gauss code of K* | O1O2O3U4U5U6O4U1U3O6O5U2 |
| Gauss code of -K* | O1O2O3U2O4O5U1U3O6U5U4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 -1 0 3 1],[ 2 0 2 1 2 2 0],[ 1 -2 0 0 2 3 1],[ 1 -1 0 0 1 2 1],[ 0 -2 -2 -1 0 2 1],[-3 -2 -3 -2 -2 0 0],[-1 0 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -2 -2 -3 -2],[-1 0 0 -1 -1 -1 0],[ 0 2 1 0 -1 -2 -2],[ 1 2 1 1 0 0 -1],[ 1 3 1 2 0 0 -2],[ 2 2 0 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,0,1,2,2,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,0,1,2,2,0,1,2] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,0,0,3,3,0,-1,1,1,0,1,2,0,1,2] |
| Phi of K* | [-3,-1,0,1,1,2,2,1,1,2,3,0,1,1,3,-1,0,0,0,-1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,2,2,0,2,0,1,1,2,2,1,3,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+8w^3z^2-6w^3z+25w^2z+15w |
| Inner characteristic polynomial | t^6+38t^4+23t^2+1 |
| Outer characteristic polynomial | t^7+54t^5+64t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -192*K1**2*K2**4 + 928*K1**2*K2**3 - 3536*K1**2*K2**2 - 64*K1**2*K2*K4 + 4544*K1**2*K2 - 3448*K1**2 + 320*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4176*K1*K2*K3 + 416*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1560*K2**4 - 656*K2**2*K3**2 - 8*K2**2*K4**2 + 1648*K2**2*K4 - 2192*K2**2 + 432*K2*K3*K5 - 1120*K3**2 - 326*K4**2 - 72*K5**2 + 2436 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |